A general chain rule for distributional derivatives
A general chain rule for distributional derivatives
We prove a general chain rule for the distribution derivatives of the composite function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon left-parenthesis x right-parenthesis equals f left-parenthesis u left-parenthesis x right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>υ<!-- υ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> …